Abstract : Hall algebras and related notions play a prominent role in the modern representation theory. In their present form, they first appeared in a series of papers by Ringel on quantum groups. After giving all the necessary definitions, I will explain the interplay between different exact structures on an additive category and degenerations of the associated Hall algebras. For the categories of representations of Dynkin quivers, this recovers degenerations of the nilpotent part of the corresponding quantum group.
To realize an entire quantum group as a version of a Hall algebra, one has to consider a more complicated category. I will explain how to recover the comultiplication of the quantum group by taking a certain unexpected exact structure on this category. If time permits, we will discuss several related conjectures.
Based on joint work and an ongoing project with Xin Fang.
[ L’exposé sera virtuel et se déroulera sur Zoom. Contacter l’organisateur LP pour obtenir les codes de connexion ].